Would you believe it...the same thing that's happening with iCloud email is now happening for me with both iMessage and SMS messages (no/intermittent notification when device is off, notification when device is on, works on the Mac) on both my iPad and iPhone both running iOS 10.1.
The message thing started for me this afternoon several hours ago. A friend of mine said he noticed it too. His is back up and running. Mine isn't.
EDIT: allegedly, the notification for messages only on the active device is a new feature.
I've started getting iMessage problems too this morning. Not coming through to the phone but seem to be ok on the iPad. WTF?!?!?
Never had so many issues.
Maybe good news, they are working on it?